Ellipticity of chains and domains

It is useful to compare the overall shapes of domains with those of proteins. A particularly useful, yet simple, method of visualising, both qualitatively and quantitatively, the overall shape and globularity of molecular structures is an ellipsoidal representation (Prabhakaran & Ponnuswamy, 1984; Taylor et al., 1983). The ellipsoidal representation of a molecule approximates an ellipsoid which has the same centre of gravity and moments of inertia as the molecule. The shape of the molecule can be essentially described by the magnitude of the three principle axes of the ellipsoid.

For a discrete system of n points, the inertial tensor, I , about the centre of mass is defined (Hughes & Gaylord, 1964) by:

I =


is the weight (or mass) associated with the point i

, , represent the position of the point i and

, ,

I is a real symmetric matrix and it is possible to calculate the moments of inertia about the principal axis by computing the eigenvalues, l, and eigenvectors, Io, of I. In matrix notation

I . Io = l . Io where l = and Io =

The eigenvalues (l1, l2, l3) represent the moments of inertia about the principal axis and the eigenvectors , and represent the direction cosines of the three orthogonal planes containing the principal axis (it should be noted that these three planes also represent the best, intermediate and worst least squares planes). The semi-axis a, b and c of an equivalent ellipsoid with moments of inertial l1, l2, l3 about its principal axis are given by the following relationships :

, ,

In practice, unit weights (i.e. = 1) were used and also since a number protein databank files contained only Ca atoms (see Table 1), for consistency, only Ca atom positions were included in the computation of I. The eigenvalues of I can be calculated using a number of standard procedures (Press et al., 1982). The semi-axis a, b and c can then be derived from (l1, l2, l3) using the above relationships.

We describe the globularity of a molecular structure by the ellipticity of the fitted inertial ellipsoid. The ellipticity, e, is defined as

e = (longest ellipsoidal axis) / (shortest elipsoidal axis)

For a sphere the ellipticity, e, is equal to 1.

Solvent Accessible Areas

It has been found that the surface area accessible to solvent of monomeric proteins is a simple function of molecular weight ( Chothia, 1976; Miller et al., 1987; Bryant et al., 1989; Islam & Weaver, 1991). We wish to extend the earlier study of Argos (1988) and investigate if there is a correlation between the solvent accessible area of individual domains and their molecular weights and if this behaviour is similar to that of single domain monomeric proteins.

Solvent accessible areas were calculated for each domain (i) as an isolated unit (Ai) and in the presence of all the polypeptide chain (Api). The program is based on the algorithm of Richmond & Richards (1978). The change in solvent accessible area for the domain, dAi, due to domain packing is given by:

dAi = Ai - Api. dA indicates the "tightness" of binding of a domain within a protein. For all the domains in a chain the total change in accessible area is given by:

DA = S dAi